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Lectures on Algebraic Groups Alexander Kleshchev Contents Part one: Algebraic Geometry page 1 1 General Algebra 3 2 Commutative Algebra 5 2.1 Some random facts 5 2.2 Ring extensions 8 3 Affine and Projective Algebraic Sets 18 3.1 Zariski topology 18 3.2 Nullstellensatz 20 3.3 Regular functions 22 3.4 Irreducible components 23 3.5 Category of algebraic sets 25 3.6 Products 28 3.7 Rational functions 29 3.8 Projective n-space 32 3.9 Functions 34 3.10 Product of projective algebraic sets 35 3.11 Example: Grassmann varieties and flag varieties 36 3.12 Example: Veronese variety 39 3.13 Problems 41 4 Varieties 46 4.1 Affine varieties 46 4.2 Prevarieties 49 4.3 Products 51 4.4 Varieties 53 4.5 Dimension 54 4.6 Problems 60 5 Morphisms 63 5.1 Fibers 63 iii iv Contents 5.2 Finite morphisms 64 5.3 Image of a morphism 66 5.4 Open and birational morphisms 69 5.5 Problems 70 6 Tangent spaces 72 6.1 Definition of tangent space 72 6.2 Simple points 74 6.3 Local ring of a simple point 76 6.4 Differential of a morphism 77 6.5 Module of differentials 78 6.6 Simple points revisited 83 6.7 Separable morphisms 85 6.8 Problems 87 7 Complete Varieties 88 7.1 Main Properties 88 7.2 Completeness of projective varieties 89 Part two: Algebraic Groups 91 8 Basic Concepts 93 8.1 Definition and first examples 93 8.2 First properties 95 8.3 Actions of Algebraic Groups 98 8.4 Linear Algebraic Groups 100 8.5 Problems 102 9 Lie algebra of an algebraic group 105 9.1 Definitions 105 9.2 Examples 107 9.3 Ad and ad 108 9.4 Properties of subgroups and subalgebras 110 9.5 Automorphisms and derivations 111 9.6 Problems 112 10 Quotients 114 10.1 Construction 114 10.2 Quotients 115 10.3 Normal subgroups 118 10.4 Problems 120 11 Semisimple and unipotent elements 121 11.1 Jordan-Chevalley decomposition 121 11.2 Unipotent algebraic groups 124 11.3 Problems 125 Contents v 12 Characteristic 0 theory 127 12.1 Correspondence between groups and Lie algebras 127 12.2 Semisimple groups and Lie algebras 129 12.3 Problems 130 13 Semisimple Lie algebras 131 13.1 Root systems 131 13.2 Semisimple Lie algebras 134 13.3 Construction of simple Lie algebras 137 13.4 Kostant Z-form 139 13.5 Weights and representations 140 13.6 Problems 142 14 The Chevalley construction 145 14.1 Definition and first properties 146 15 Borel subgroups and flag varieties 148 15.1 Complete varieties and Borel’s fixed point theorem 148 15.2 Borel subgroups 149 15.3 The Bruhat order 151 16 The classification of reductive algebraic groups 154 16.1 Maximal tori and the root system 154 16.2 Sketch of the classification 156 Bibliography 160 Part one Algebraic Geometry 1 General Algebra Definition 1.0.1 A functor F : A → B is called faithful if the map HomA(A1,A2) → HomB(F (A1),F (A2)), θ 7→ F (θ) (1.1) is injective, and F is called full if the map (1.1) is surjective. Theorem 1.0.2 A functor F : A → B is an equivalence of categories if and only if the following two conditions hold: (i) F is full and faithful; (ii) every object of B is isomorphic to an object of the form F (A) for some A ∈ Ob A. Proof ( ⇒ ) Let F be an equivalence of categories and G : B → A be the quasi-inverse functor. Let α : GF → idA and β : FG → idB be isomorphisms of functors. First of all, for any object B of B βB : F (G(B)) → B is an isomorphism, which gives (ii). Next, for each ϕ ∈ HomA(A1,A2) we have the commutative diagram α A-1 GF (A1) A1 GF (ϕ) ϕ ? α ? A-2 GF (A2) A2 Hence ϕ can be recovered from F (ϕ) by the formula −1 ϕ = αA2 ◦ GF (ϕ) ◦ (αA1 ) . (1.2) This shows that F is faithful. Similarly, G is faithful. To prove that F 3 4 General Algebra is full, consider an arbitrary morphism ψ ∈ HomB(F (A1),F (A2)), and set −1 ϕ := αA2 ◦ G(ψ) ◦ (αA1 ) ∈ HomA(A1,A2). Comparing this with (1.2) and taking into account that αA1 and αA2 are isomorphisms, we deduce that G(ψ) = GF (ϕ). As G is faithful, this implies that ψ = F (ϕ), which completes the proof that F is a full functor. ( ⇐ ) Assume that (i) and (ii) hold. In view of (i), we can (and will) identify the set HomB(F (A1),F (A2)) with the set HomA(A1,A2) for any A1,A2 ∈ Ob A. Using (ii), for each object B in B we can pick an object AB in A and an isomorphism βB : F (AB) → B. We define a functor G : B → A which will turn out to be a quasi-inverse functor to F . on the objects we set G(B) = AB for any B ∈ Ob B. To define G on the morphisms, let ψ ∈ HomB(B1,B2). G(ψ) := β−1 ◦ ψ ◦ β ∈Hom (FG(B ),FG(B )) B2 B1 B 1 2 = HomA(G(B1),G(B2)). It is easy to see that G is a functor, and β = {βB} : FG → idB is an isomorphism of functors. Further, βF (A) = F (αA) for the unique morphism αA : GF (A) → A. Finally, it is not hard to see that α = {αA} : GF → idA is an isomorphism of functors. 2 Commutative Algebra Here we collect some theorems from commutative algebra which are not always covered in 600 algebra. All rings and algebras are assumed to be commutative. 2.1 Some random facts Lemma 2.1.1 Let k be a field, f, g ∈ k[x, y], and assume that f is irreducible. If g is not divisible by f, then the system f(x, y) = g(x, y) = 0 has only finitely many solutions. Proof See [Sh, 1.1]. Proposition 2.1.2 Let A, B be k-algebras, I / A, J / B be ideals. Then ¯ A/I ⊗k B/J → (A ⊗k B)/(A ⊗ J + I ⊗ B), a¯ ⊗ b 7→ a ⊗ b is an isomorphism of algebras. Definition 2.1.3 A subset S of a commutative ring R is called multi- plicative if 1 ∈ S and s1s2 ∈ S whenever s1, s2 ∈ S. A multiplicative subset is called proper if 0 6∈ S. Lemma 2.1.4 Let S ⊂ R be a proper multiplicative set. Let I be an ideal of R satisfying I ∩ S = ∅. The set T of ideals J ⊇ I such that J ∩ S = ∅ has maximal elements, and each maximal element in T is a prime ideal. 5 6 Commutative Algebra Proof That the set T has maximal elements follows from Zorn Lemma. Let M be such an element. Assume that x, y ∈ R \ M. By the choice of M, M + Rx contains some s1 ∈ S and M + Ry contains some s2 ∈ S, i.e. s1 = m1 + r1x and s2 = m2 + r2y. Hence s1s2 = (m1 + r1x)(m2 + r2y) ∈ M + Rxy. It follws that M + Rxy 6= M, i.e. xy 6∈ M. Theorem 2.1.5 (Prime Avoidance Theorem) Let P1,...,Pn be prime ideals of the ring R. If some ideal I is contained in the union P1 ∪ · · · ∪ PN , then I is already contained in some Pi. Proof We can assume that none of the prime ideals is contained in another, because then we could omit it. Fix an i0 ∈ {1,...,N} and for each i 6= i0 choose an fi ∈ Pi, fi 6∈ Pi0 , and choose an fi0 ∈ I, fi0 6∈ Pi0 . Q Then hi0 := fi lies in each Pi with i 6= i0 and I but not in Pi0 . Now, P hi lies in I but not in any Pi. Lemma 2.1.6 (Nakayama’s Lemma) Let M be a finitely generated module over the ring A. Let I be an ideal in A such that for any a ∈ 1+I, aM = 0 implies M = 0. Then IM = M implies M = 0. Proof Let m1, . , ml be generators of M. The condition IM = M means that l X mi = xijmj (1 ≤ i ≤ l). j=1 for some xij ∈ I. Hence l X (xij − δij)mj = 0 (1 ≤ i ≤ l). j=1 So by Cramer’s rule, dmj = 0, where d = det(xij − δij). Hence dM = 0. But d ∈ 1 + I, so M = 0. Corollary 2.1.7 If B ⊃ A is a ring extension, and B is finitely gener- ated as an A-module, then IB 6= B for any proper ideal I of A. Proof Since B contains 1, we have aB = 0 only if a = 0. Now all elements of 1 + I are non-zero for a proper ideal I, so we can apply Nakayama’s Lemma. 2.1 Some random facts 7 Corollary 2.1.8 (Nakayama’s Lemma) Let M be a finitely generated module over the ring A, M 0 ⊆ M be a submodule, and let I be an ideal in A such that all elements of 1 + I are invertible. Then IM + M 0 = M implies M 0 = M. Proof Apply Lemma 2.1.6 to M/M 0. Another version: Corollary 2.1.9 (Nakayama’s Lemma) Let M be a finitely generated module over a ring A, and I be a maximal ideal of A. If IM = M, then there exists x 6∈ M such that xM = 0.
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